Socializing neutered vs. non-neutered stray - Help!

I rescued a stray about 10 days ago.  I've been working with her and she shows some promise although I believe she is about 85% feral at this time.  She will not look me in the eye.  I have not touched her.  I'm trying to arrange spaying (pretty sure it's a she) with the ASPCA or Humane Society and am finding myself falling through the cracks of their categories. Should I push to get this done sooner than later or should I hold on for 4-6 weeks before I can get an appt.  Does anyone have an opinion about whether it is easier to socialize a spayed cat vs. an intact cat?   One consideration is that I am going to have to trap her again in order to get her to the vet. That will be traumatic for both of us. I know it will set back the socialization.  I need all the advice I can gather right now.  

FYI I have three cats of my own, one is 19 years old. I cannot take this new girl myself although I would if I could.

She should definitely be spayed sooner rather than later, especially if she has access to the outdoors. She could be pregnant already, and it's easier to spay earlier in the pregnancy. Taking a cat to the vet will usually set back some progress, but I think it's better to do that early on so if you have to start from the beginning afterwards, at least you'll only have lost a few days rather than several weeks of work.. Cats also feel better after they've been fixed and are getting any medication they might need, and it helps that they can associate you with feeling better.

Regardless of when she get spayed, I think she should be trapped and kept indoors if you want to socialize her. A bathroom or even a large crate is fine to keep her in, she won't want more room than that anyway in the beginning. If she's not in a cage, give her some places to hide like a carrier or cardboard boxes so she can feel more secure. This is a very detailed guide for socializing cats: bestfriends.org/resources/socializing-very-shy-or-fearful-cats

Definitely get her spayed ASAP.  There may actually be a chance that she is already pregnant. 

As for getting her to the vet, you can start feeding her either in the trap or in a larger dog crate or dog carrier.  When I brought one of my feral boys inside the house, I used a large carrier that zipped up at the front.  I kept it all the way open and fed the cat in the carrier for every meal.  On the morning of the vet appointment, I put the food inside, he went inside and I blocked the exit with a piece of dry wall.  Then I zipped the carrier shut and off we went.  It was much easier than re trapping.

First, thank you so much for replies.  I should have supplied more details in my post.  I have already captured this cat.  She has been living in a basement storage area of a building in my complex for more than three months, fed and with litter box by someone else.  I currently have her on an enclosed terrace of a friend's apartment.  I sit with her every day but especially while she's eating.  She does not get food unless I am there, sitting on the floor.  

I have to move her to a new foster apt. before the end of April, which means I have to trap her again.  I hate that.  I was hoping to trap, get spayed then release in her new digs but it appears that is not going to happen.  I'm still trying to find help with getting her spayed.  I am on Medicare so I can do this economically at ASPCA or the Humane Society.  But those places have not been helpful so far.  They talk about bringing your "pet."  Well, this girl is no one's pet yet.  There's a 4-6 week wait for an appt. with the Humane Society.  I'm actually waiting for a call back from them.  But honestly, I don't think this is going to happen soon.  If I can find a TNR group to help me, then I might have a chance of getting it done soon.  I'm in Manhattan.  If anyone out there is part of a TNR group that can help me with this girl, that would be more awesome than I can express.  I may just have to move her and continue to try to socialize her while I wait for an appt.  The new digs are also only temporary. 

So, she eats right beside me.  This morning she was pushing the can right next to my hand.  But then she backs off.  I have not tried to touch her.  All advice welcome.  Love to you angels.